Linear and Multilinear Algebra, Vol. 56, Nos. 1–2, Jan.–Mar. 2008, 105–130 Ranks and determinants of the sum of matrices from unitary orbits CHI-KWONG LI*y, YIU-TUNG POONz and NUNG-SING SZEx yDepartment of Mathematics, College of William & Mary, Williamsburg, VA 23185, USA zDepartment of Mathematics, Iowa State University, Ames, IA 50011, USA xDepartment of Mathematics, University of Connecticut, Storrs, CT 06269, USA Communicated by C. Tretter (Received 30 January 2007; in final form 27 March 2007) The unitary orbit UðAÞ of an nÂn complex matrix A is the set consisting of matrices unitarily similar to A. Given two nÂn complex matrices A and B, ranks and determinants of matrices of the form X þ Y with ðX, Y Þ2UðAÞUðBÞ are studied. In particular, a lower bound and the best upper bound of the set RðA, BÞ¼frankðX þ YÞ : X2UðAÞ, Y2UðBÞg are determined. It is shown that ÁðA, BÞ¼fdetðX þ YÞ : X2UðAÞ, Y2UðBÞg has empty interior if and only if the set is a line segment or a point; the algebraic structure of matrix pairs (A, B) with such properties are described. Other properties of the sets R(A, B) and ÁðA, BÞ are obtained. The results generalize those of other authors, and answer some open problems. Extensions of the results to the sum of three or more matrices from given unitary orbits are also considered. Downloaded By: [Li, Chi-Kwong] At: 00:07 20 March 2009 Keywords: Rank; Determinant; Matrices; Unitary orbit 2000 Mathematics Subject Classifications: 15A03; 15A15 1. Introduction Let A2Mn. The unitary orbit of A is denoted by à à UðAÞ¼fgUAU : U U ¼ In : Evidently, if A is regarded as a linear operator acting on Cn, then UðAÞ consists of the matrix representations of the same linear operator under different *Corresponding author. Email: [email protected] Dedicated to Professor Yik-Hoi Au-Yeung on the occasion of his 70th birthday. Linear and Multilinear Algebra ISSN 0308-1087 print/ISSN 1563-5139 online ß 2008 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/03081080701369306 106 C.-K. Li et al. orthonormal bases. Naturally, UðAÞ captures many important features of the operator A. For instance, A is normal if and only if UðAÞ has a diagonal matrix; A is Hermitian (positive semi-definite) if and only if UðAÞ contains a (nonnegative) real diag- onal matrix; A is unitary if and only if UðAÞ has a diagonal matrix with unimodular diagonal entries. There are also results on the characterization of diagonal entries and submatrices of matrices in UðAÞ; see [14,20,23,30] and their references. In addition, the unitary orbit of A has a lot of interesting geometrical and algebraic properties, see [11]. Motivated by theory as well as applied problems, there has been a great deal of interest in studying the sum of two matrices from specific unitary orbits. For example, eigenvalues of UAUà þ VBVà for Hermitian matrices A and B were completely determined in terms of those of A and B (see [16] and its references); the optimal norm estimate of UAUà þ VBVà was obtained (see [10] and its references); à à the range of values of detðUAU þ VBV Þ for Hermitian matrices A, B2Mn was described, see [15]. Later, Marcus and Oliveira [26,29] conjectured that if A, B2Mn are normal matrices with eigenvalues a1, ..., an and b1, ..., bn, respectively, then for à à any unitary U, V2Mn, detðUAU þ VBV Þ always lies in the convex hull of () Yn ÀÁ PðA, BÞ¼ aj þ bðjÞ : is a permutation of f1, ..., ng : ð1:1Þ j¼1 In connection to this conjecture, researchers [2–5,7,8,12,13] considered the deter- minantal range of A, B2Mn defined by ÁðA, BÞ¼fdetðA þ UBUÃÞ: U is unitaryg: This can be viewed as an analog of the generalized numerical range of A and B defined by WðA, BÞ¼ftrðAUBUÃÞ: U is unitaryg, Downloaded By: [Li, Chi-Kwong] At: 00:07 20 March 2009 which is a useful concept in pure and applied areas; see [18,21,25] and their references. In this article, we study some basic properties of the matrices in UðAÞþUðBÞ¼fX þ Y: ðX, Y Þ2UðAÞUðBÞg: The focus will be on the rank and the determinant of these matrices. Our article is organized as follows. In section 2, we obtain a lower bound and the best upper bound for the set RðA, BÞ¼frankðX þ Y Þ: X 2UðAÞ, Y 2UðBÞg; moreover, we characterize those matrix pairs (A, B) such that R(A, B) is a singleton, and show that the set R(A, B) has the form fk, k þ 1, ..., ng if A and ÀB have no common eigenvalues. On the contrary if A and ÀB are orthogonal projections, then the rank values of matrices in UðAÞþUðBÞ will either be all even or all odd. In section 3, we characterize matrix pairs ðA, BÞ such that ÁðA, BÞ has empty interior, which is Ranks and determinants of the sum of matrices 107 possible only when ÁðA, BÞ is a singleton or a nondegenerate line segment. This extends the results of other researchers who treated the case when A and B are normal [2–4,9]. In particular, our result shows that it is possible to have normal matrices A and B such that ÁðA, BÞ is a subset of a line, which does not pass through the origin. This disproves a conjecture in [9]. In [3], the authors showed that if A, B2Mn are normal matrices such that the union of the spectra of A and ÀB consists of 2n distinct elements, then every nonzero sharp point of ÁðA, BÞ is an element in P(A, B). (See the definition of sharp point in section 3.) We showed that every (zero or nonzero) sharp point of ÁðA, BÞ belongs to P(A, B) for arbitrary matrices A, B2Mn. In section 4, we consider the sum of three or more matrices from given unitary orbits, and matrix orbits corresponding to other equivalence relations. In the literature, some authors considered the set DðA, BÞ¼fdetðX À YÞ: X 2UðAÞ, Y 2UðBÞg instead of ÁðA, BÞ. Evidently, we have DðA, BÞ¼ÁðA, ÀBÞ. It is easy to translate results on D(A, B) to those on ÁðA, BÞ, and vice versa. Indeed, for certain results and proofs, it is more convenient to use the formulation of D(A, B). We will do that in section 3. On the other hand, it is more natural to use the summation formulation to discuss the extension of the results to matrices from three or more unitary orbits. 2. Ranks 2.1. Maximum and minimum rank In [10], the authors obtained optimal norm bounds for matrices in UðAÞþUðBÞ for two given matrices A, B2Mn. By the triangle inequality, we have à à maxfkUA U þ VBV k: U, V unitarygminfkA À InkþkB þ Ink: 2 Cg: Downloaded By: [Li, Chi-Kwong] At: 00:07 20 March 2009 It was shown in [10] that the inequality is actually an equality. For the rank functions, we have à à maxfrankðUAU þ VBV Þ: U,V unitarygminfrankðA À InÞþrankðB þ InÞ: 2 Cg: Of course, the right side may be strictly larger than n, and thus equality may not hold in general. It turns out that this obstacle can be overcome easily as shown in the following. THEOREM 2.1 Let A, B2Mn and m ¼ minfrankðA À InÞþrankðB þ InÞ: 2 Cg ¼ minfrankðA À InÞþrankðB þ InÞ: is an eigenvalue of A ÈÀBg: Then maxfrankðUAUà þ VBVÃÞ : U, V unitaryg¼minfm, ng: 108 C.-K. Li et al. Proof If is an eigenvalue of A ÈÀB, then rankðA À InÞþrankðB þ InÞ2n À 1; if is not an eigenvalue of A ÈÀB, then rankðA À InÞþrankðB þ InÞ¼2n: As a result, rankðA À InÞþrankðB þ InÞ will attain its minimum at an eigenvalue of the matrix A ÈÀB. It is clear that maxfrankðUA U à þ VBVÃÞ: U, V unitarygminfm, ng: It remains to show that there are U, V such that UAUà þ VBVà has rank equal to minfm, ng. Suppose mn and there is such that rankðA À InÞ¼k and rankðB þ InÞ¼m À k. We may replace (A, B)byðA À In, B þ InÞ and assume that ¼ 0. Furthermore, we may assume that km À k; otherwise, interchange A and B. Let A ¼ XDY be such that X, Y2Mn are unitary, and D ¼ D1 È 0nÀk with invertible diagonal D1. Replace A by YAY*, we may assume that U11 U12 D1 0 A ¼ UD ¼ U21 U22 00nÀk with U ¼ YX. Similarly, we may assume that V11 V12 0k 0 B ¼ VE ¼ , V21 V22 0 E2 where V is a unitary matrix and E2 is a diagonal matrix with rank m À k. Let W be a unitary matrix such that the first k columns of WU together with the last n À k columns of V are linearly independent. That is, if W11 W12 W ¼ , Downloaded By: [Li, Chi-Kwong] At: 00:07 20 March 2009 W21 W22 the matrix W11U11 þ W12U21 V12 W21U11 þ W22U21 V22 à is invertible. If W11 is invertible, then D1W11 has rank k and so à à W11U11 þ W12U21 V12 D1W D1W WAWà þ B ¼ 11 21 W21U11 þ W22U21 V22 0 E2 has rank m.IfW11 is not invertible, we will replace W by W~ obtained as follows. By the CS decomposition, there are unitary matrices P1, Q1 2Mk and P2, Q2 2MnÀk such that CS ðP1 È P2ÞWðQ1 È Q2Þ¼ È InÀ2k, ÀSC Ranks and determinants of the sum of matrices 109 qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 where C ¼ diagðc1, ..., ckÞ with 1 c1 ÁÁÁck 0 and S ¼ diagð 1 À c1, ..., 1 À ck Þ.